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C22×C3⋊ΓL(1,𝔽16) #10624355

order720
isoclinismisoclinic class of C3⋊ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC22×C3⋊ΓL(1,𝔽16)
unambiguous semidirect productD5⋊(C2×C3⋊Dic3)

Splittings

Fitting, socleC22×C32×C5 ⋊ C4
derived, HallC32×C5 ⋊ C22×C4
SylowC32 ⋊ C22×AGL(1,𝔽5)
SylowC5 ⋊ C22×C3⋊Dic3

Subgroups

subgroupG/H
center, hypercenterC22C3⋊ΓL(1,𝔽16)
commutator, hypocenterC32×C5C22×C4
FrattiniC1G
Fitting, socleC22×C32×C5C4
prime radicalC22×C32×D5C2
cosocleC32×D5C23
LCSG ▷ C32×C5
UCSC1 ◁ C22

324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 180

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽5))⋊(C3⋊Dic3), #13085462, #13086308, …
maximal normal subgroup of(C2×AGL(1,𝔽5))⋊(C3⋊Dic3), #13085462, #13086308, …
maximal quotient ofC2×D5⋊(C3⋊(C3⋊C8)), C2×#10624093, D5⋊#3748, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 824);